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Register a new userComplementary Ramsey numbers and Ramsey graphs
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In this paper, we consider a variant of Ramsey numbers which we call complementary Ramsey numbers $bar{R}(m,t,s)$. We first establish their connections to pairs of Ramsey $(s,t)$-graphs. Using the classification of Ramsey $(s,t)$-graphs for small $s,t$, we determine the complementary Ramsey numbers $bar{R}(m,t,s)$ for $(s,t)=(4,4)$ and $(3,6)$.
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Given graphs $H_1, dots, H_t$, a graph $G$ is $(H_1, dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $iin{1, dots, t}$, but any proper subgraph of $G $ does not possess this property. We define $mathcal{R}_{min}(H_1, dots, H_t)$ to be the family of $(H_1, dots, H_t)$-Ramsey-minimal graphs. A graph $G$ is dfn{$mathcal{R}_{min}(H_1, dots, H_t)$-saturated} if no element of $mathcal{R}_{min}(H_1, dots, H_t)$ is a subgraph of $G$, but for any edge $e$ in $overline{G}$, some element of $mathcal{R}_{min}(H_1, dots, H_t)$ is a subgraph of $G + e$. We define $sat(n, mathcal{R}_{min}(H_1, dots, H_t))$ to be the minimum number of edges over all $mathcal{R}_{min}(H_1, dots, H_t)$-saturated graphs on $n$ vertices. In 1987, Hanson and Toft conjectured that $sat(n, mathcal{R}_{min}(K_{k_1}, dots, K_{k_t}) )= (r - 2)(n - r + 2)+binom{r - 2}{2} $ for $n ge r$, where $r=r(K_{k_1}, dots, K_{k_t})$ is the classical Ramsey number for complete graphs. The first non-trivial case of Hanson and Tofts conjecture for sufficiently large $n$ was setteled in 2011, and is so far the only settled case. Motivated by Hanson and Tofts conjecture, we study the minimum number of edges over all $mathcal{R}_{min}(K_3, mathcal{T}_k)$-saturated graphs on $n$ vertices, where $mathcal{T}_k$ is the family of all trees on $k$ vertices. We show that for $n ge 18$, $sat(n, mathcal{R}_{min}(K_3, mathcal{T}_4)) =lfloor {5n}/{2}rfloor$. For $k ge 5$ and $n ge 2k + (lceil k/2 rceil +1) lceil k/2 rceil -2$, we obtain an asymptotic bound for $sat(n, mathcal{R}_{min}(K_3, mathcal{T}_k))$.
We estimate Ramsey numbers for bipartite graphs with small bandwidth and bounded maximum degree. In particular we determine asymptotically the two and three color Ramsey numbers for grid graphs. More generally, we determine asymptotically the two color Ramsey number for bipartite graphs with small bandwidth and bounded maximum degree and the three color Ramsey number for such graphs with the additional assumption that the bipartite graph is balanced.
Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.
For an integer $qge 2$, a graph $G$ is called $q$-Ramsey for a graph $H$ if every $q$-colouring of the edges of $G$ contains a monochromatic copy of $H$. If $G$ is $q$-Ramsey for $H$, yet no proper subgraph of $G$ has this property then $G$ is called $q$-Ramsey-minimal for $H$. Generalising a statement by Burr, Nev{s}etv{r}il and Rodl from 1977 we prove that, for $qge 3$, if $G$ is a graph that is not $q$-Ramsey for some graph $H$ then $G$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$, as long as $H$ is $3$-connected or isomorphic to the triangle. For such $H$, the following are some consequences. (1) For $2le r< q$, every $r$-Ramsey-minimal graph for $H$ is contained as an induced subgraph in an infinite number of $q$-Ramsey-minimal graphs for $H$. (2) For every $qge 3$, there are $q$-Ramsey-minimal graphs for $H$ of arbitrarily large maximum degree, genus, and chromatic number. (3) The collection ${{cal M}_q(H) : H text{ is 3-connected or } K_3}$ forms an antichain with respect to the subset relation, where ${cal M}_q(H)$ denotes the set of all graphs that are $q$-Ramsey-minimal for $H$. We also address the question which pairs of graphs satisfy ${cal M}_q(H_1)={cal M}_q(H_2)$, in which case $H_1$ and $H_2$ are called $q$-equivalent. We show that two graphs $H_1$ and $H_2$ are $q$-equivalent for even $q$ if they are $2$-equivalent, and that in general $q$-equivalence for some $qge 3$ does not necessarily imply $2$-equivalence. Finally we indicate that for connected graphs this implication may hold: Results by Nev{s}etv{r}il and Rodl and by Fox, Grinshpun, Liebenau, Person and Szabo imply that the complete graph is not $2$-equivalent to any other connected graph. We prove that this is the case for an arbitrary number of colours.
A set of vertices $Xsubseteq V$ in a simple graph $G(V,E)$ is irredundant if each vertex $xin X$ is either isolated in the induced subgraph $langle Xrangle$ or else has a private neighbor $yin Vsetminus X$ that is adjacent to $x$ and to no other vertex of $X$. The emph{irredundant Ramsey number} $s(m,n)$ is the smallest $N$ such that in every red-blue coloring of the edges of the complete graph of order $N$, either the blue subgraph contains an $m$-element irredundant set or the red subgraph contains an $n$-element irredundant set. The emph{mixed Ramsey number} $t(m,n)$ is the smallest $N$ for which every red-blue coloring of the edges of $K_N$ yields an $m$-element irredundant set in the blue subgraph or an $n$-element independent set in the red subgraph. In this paper, we first improve the upper bound of $t(3,n)$; using this result, we confirm that a conjecture proposed by Chen, Hattingh, and Rousseau, that is, $lim_{nrightarrow infty}frac{t(m,n)}{r(m,n)}=0$ for each fixed $mgeq 3$, is true for $mleq 4$. At last, we prove that $s(3,9)$ and $t(3,9)$ are both equal to $26$.
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